Question

The quarterly returns for a group of 51 mutual funds are well modeled by a Normal model with a mean of 7.4​% and a standard deviation of 3.4​%. Use the​ 68-95-99.7 Rule to find the cutoff values that would separate the following percentages of​ funds, rather than using technology to find the exact values. ​a) the highest 50​% ​b) the highest 16​% ​c) the lowest 2.5​% ​d) the middle 68​% ​a) Select the correct choice and fill in any answer boxes in your choice below. ​(Round to one decimal place as​ needed.) A. xless than or equals 50​% B. xgreater than or equals nothing​% C. nothing​%less than or equalsxless than or equals nothing​% ​b) Select the correct choice and fill in any answer boxes in your choice below. ​(Round to one decimal place as​ needed.) A. xgreater than or equals nothing​% B. xless than or equals nothing​% C. nothing​%less than or equalsxless than or equals nothing​% ​c) Select the correct choice and fill in any answer boxes in your choice below. ​(Round to one decimal place as​ needed.) A. xless than or equals nothing​% B. xgreater than or equals nothing​% C. nothing​%less than or equalsxless than or equals nothing​% ​d) Select the correct choice and fill in any answer boxes in your choice below. ​(Round to one decimal place as​ needed.)

Answer 1

Answer:

A) x ≥ 0.074; B) x ≥ 0.108; C) x ≤ 0.006; D) 0.04 ≤ x ≤ 0.108

Step-by-step explanation:

68% of data will fall within 1 standard deviation of the mean; 95% of data will fall within 2 standard deviations of the mean; and 99.7% of data will fall within 3 standard deviations of the mean.

Breaking this down, we find that 34% of data fall from the mean to 1 standard deviation above the mean; 13.5% of data fall from 1 standard deviation above the mean to 2 standard deviations above the mean; 2.35% of data fall from 2 standard deviations above the mean to 3 standard deviations above the mean; and 0.15% of data fall above 3 standard deviations above the mean.

The same percentages apply to the standard deviations below the mean.

The highest 50% of data will fall from the mean to the end of the right tail.  This means the inequality for the highest 50% will be x ≥ 0.074, the mean.

The highest 16% of data will fall from 1 standard deviation above the mean to the end of the right tail.  This means the inequality for the highest 16% will be x ≥ 0.074+0.034, or x ≥ 0.108.

The lowest 2.5% of data will fall from 2 standard deviations below the mean to the end of the left tail.  This means the inequality for the lowest 2.5% will be x ≤ 0.074-0.034-0.034, or x ≤ 0.066.

The middle 68% will fall from 1 standard deviation below the mean to 1 standard deviation above the mean; this means the inequality for the middle 68% will be

0.074-0.034 ≤ x ≤ 0.074+0.034, or

0.04 ≤ x ≤ 0.108